prove that √5+√3 is irrational
Answers
Answer:
To prove :
3
+
5
is irrational.
Let us assume it to be a rational number.
Rational numbers are the ones that can be expressed in
q
p
form where p,q are integers and q isn't equal to zero.
3
+
5
=
q
p
3
=
q
p
−
5
squaring on both sides,
3=
q
2
p
2
−2.
5
(
q
p
)+5
⇒
q
(2
5
p)
=5−3+(
q
2
p
2
)
⇒
q
(2
5
p)
=
q
2
2q
2
−p
2
⇒
5
=
q
2
2q
2
−p
2
.
2p
q
⇒
5
=
2pq
(2q
2
−p
2
)
As p and q are integers RHS is also rational.
As RHS is rational LHS is also rational i.e
5
is rational.
But this contradicts the fact that
5
is irrational.
This contradiction arose because of our false assumption.
so,
3
+
5
irrational.
Step-by-step explanation:
Given√3 + √5
To prove:√3 + √5 is an irrational number.
Let us assume that√3 + √5 is a rational number.
So it can be written in the form a/b
√3 + √5 = a/b
Here a and b are coprime numbers and b ≠ 0
Solving
√3 + √5 = a/b
On squaring both sides we get,
(√3 + √5)² = (a/b)²
√3² + √5² + 2(√5)(√3) = a²/b²
3 + 5 + 2√15 = a²/b²
8 + 2√15 = a²/b²
2√15 = a²/b² – 8
√15 = (a²- 8b²)/2b
a, b are integers then (a²-8b²)/2b is a rational number.
Then √15 is also a rational number.
But this contradicts the fact that √15 is an irrational number.
Our assumption is incorrect
√3 + √5 is an irrational number.
Hence, proved.